We use the Barahona and Nenes cirrus formation parameterization (BN09) () and its adjoint (). BN09 describes the competition for water vapor between heterogeneous and homogeneous nucleation; the number of heterogeneously formed crystals, Ni,het, is calculated from any of a variety of nucleation spectra, and homogeneously formed number, Ni,hom, is calculated with an approximate solution to the coupled mass and energy balances of a cirrus cloud parcel. Then the total ice crystal number, Ni, is the sum of the heterogeneous and homogeneous contributions (Eq. ). When the temperature is greater than about -38 ∘C, homogeneous nucleation ceases because it is kinetically unfavorable. Homogeneous nucleation is also suppressed when the number of INP exceeds a certain threshold, Nlim, and the maximum supersaturation that develops within the cloud parcel, smax, is less than the threshold for homogeneous nucleation, shom. In this case, smax must be numerically calculated from the growth and supersaturation evolution equations.

Ni=Ni,hom+Ni,het(shom),Ni,het(shom)<NlimNi,het(smax),Ni,het(shom)≥Nlim

The BN parameterization make two principal assumptions: first, ice crystal growth occurs mostly in the free growth regime where new nucleation does not significantly change the parcel supersaturation; second, Ni is calculated at the maximum supersaturation rather than a supersaturation later in the freezing pulse. These assumptions lead to overestimation of Ni at lower temperatures and higher updraft velocities and underestimation of Ni at lower updrafts, when the freezing pulse is longer. For a wide range of cirrus formation conditions, however, the parameterization output matches that of a detailed parcel model to within 5 %. These points are discussed in .

As in , the TAPENADE automatic differentiation tool was used to create an adjoint model of BN09 (ABN15 hereafter) (). For the finite series of operations that link Ni to the inputs in BN09, TAPENADE uses the chain rule to propagate a perturbation in the output, dNi, back to differentials in the input variables. Once developed, the adjoint model saves significant computational time, relative to a finite difference method, and avoids both approximation and truncation errors. ABN15 differentiates Ni with respect to 13 input variables: temperature; updraft velocity; accumulation- and coarse-mode dust numbers and diameters; organic aerosol number and diameter; black carbon number and diameter; sulfate number and diameter; and water vapor deposition coefficient. All derivatives, along with the typical output of BN09, are evaluated at the input model state for each grid cell and time step of a GCM (global climate model) run. ABN15 is verified with centered finite difference approximations, using perturbations of ±0.1 % around each input for simulation-relevant thermodynamic and aerosol conditions. Such finite difference calculations require two runs for each variable, so for 13 input variables, the adjoint model saves 25 executions of the parameterization relative to typical sensitivity calculations.

Simulation inputs are generated from the NCAR Community Atmosphere Model, version 5 (CAM5) at the 232 hPa pressure level with 2-year spin-up and 2.5 × 1.88∘ resolution. The input updraft velocity from CAM 5.1 is calculated from the turbulent kinetic energy in the moist turbulence scheme of Bretherton and Park 2009 as wsub=23TKE. The probability distribution of these values is compared to 2 years' worth of millimeter cloud radar measurements (MMCR) in Figs. S1 and S2 in the Supplement with all values at the same latitude, longitude and altitude. Measurements are shown after Doppler velocity decomposition, as described in .

The distribution of hourly averaged measurements has a lower maximum and decays to smaller values than that of the hourly averaged simulation inputs. Comparing updraft distribution from aircraft and ground-based MMCR, note similar behavior in which the MMCR velocities were repeatedly smaller than the in situ ones; however, saw that lower resolution models tend to decay to even smaller values than the MMCR observations because they do not resolve the gravity wave contribution. This difference is probably due to the filtering of deep convective systems within the MMCR data but no analogous filter for simulated updrafts in this case.

We use daily averaged updraft values for which the distribution agrees better with the observed values. wsub values from CAM are used as the standard deviation σsub,w of a Gaussian updraft velocity distribution P(w) of mean μsub,w=0.1 cm s-1. Both output ice crystal numbers and sensitivities are weighted over this distribution to account for sub-grid variability (): f(w)‾=∫0∞f(w)P(w)dw∫0∞P(w)dw.

This integration is performed numerically with a six-point Legendre–Gauss quadrature method, with weights and abscissae chosen over an interval from minimum to maximum velocity, which are taken as 4 standard deviations below and above μsub,w. An upper bound of 3 m s-1, unlike that of 0.2 m s-1 used in and , and a lower bound of 0.001 m s-1 are enforced.

The altitude examined is in the middle of the cirrus cloud classification from the International Cloud Climatology Project (pressures between 440 and 50 hPa) and represents pure ice cloud formation. The emissions inventory () and MAM3 module were used (). Lognormal size distributions are assumed for all aerosol types with geometric standard deviations, σg, assumed to be constant and listed below in Table . σg and total aerosol mass are used to determine geometric mean diameter for each mode. Total aerosol number is scaled by mass fraction to determine aerosol number concentrations in each mode (). For calculations of ice crystal number concentrations and sensitivities, BN09 and ABN15 were run over a year with four heterogeneous INP spectra and daily averaged values of CAM output.

PDA08 uses the exponential correlation of crystal number and supersaturation in as a reference spectrum, extending the applicable ranges of temperature and supersaturation and incorporating characteristics of the precursor aerosol. The number of ice-nucleating particles, nINP,X in aerosol group X (dust and metallics – DM, black carbon – BC, or organics – O) is calculated with a sum over the aerosol size distribution weighted by a freezing fraction: nINP,X=∫log0.1μm∞{1-exp[-μX(D,Si,T)]}nX(logD)dlogD. μX represents the number of ice embryos forming per aerosol and is the product of the active site density and aerosol surface area (): μX=HX(Si,T)ξ(T)αXnINP,*ΩX,*πD2. nINP,* is the INP number from a reference activity spectrum; ΩX,* is a reference aerosol surface area, which acts as a normalization factor for the size distribution; αX is the portion of aerosol number belonging to group X within nINP,*; nX(logD) is the aerosol size distribution; and HX is a threshold function that reduces INP concentrations at conditions subsaturated with respect to water and warm sub-zero temperatures in agreement with observations. HX equals unity at water saturation and steps at certain threshold temperatures, T0,X, and supersaturations, si,0,X, for the different aerosol groups. Finally ξ(T) diminishes heterogeneous nucleation at warm sub-zero temperatures.

Both PDA08 and PDA13 adopt the mathematical framework of Eq. (), but PDA13 employs more extensive field campaign data (). The organic classification in PDA13 is also split into primary biological material and glassy organics, following recent observations of distinct ice-nucleating activity for these particle types. In this study, sensitivity of Ni to biological INP is not considered, as CAM5 does not currently output a biological particle number.

We also use the classical nucleation spectrum developed by and presented in conjunction with the parameterization (): nINP,X=eXnX(logD)minsisi,0,Xe-f(cosθ)khom(si,0,X-si),1, where eX is the nucleation efficiency of aerosol group X, si,0,X is the threshold supersaturation, nX(logD) is the aerosol size distribution, θ is the INP-ice contact angle, and khom is a parameter related to the homogeneous nucleation threshold. Dust and black carbon groups are included with parameters listed in Table ; contact angles come from the laboratory data of and eDM is similar to that in . The stochastic component of the nucleation efficiency through heterogeneous nucleation rate coefficient is assumed to be negligible, and the singular paradigm also underlies this spectrum. eX is potentially a function of temperature and the aerosol profile, but here it is taken from literature and assumed to be constant throughout the simulation.

The nucleation efficiency of hematite particles was measured at the AIDA chamber from -78 up to -36 ∘C and parameterized (). The third-order polynomial fit for active site density (in m-2) is given in Eq. () as a function of temperature and saturation ratio of ice. Isolines from AIDA expansion cooling experiments are interpolated over the temperature-supersaturation space, assuming a hematite baseline surface area of 6.3×10-10 m2L-1. ns(T,Si)=-3.777×1013-7.818×1011T+4.252×1013Si-4.598×109T2+6.952×1011T⋅Si-1.111×1013Si2-2.966×106T3+2.135×109T2⋅Si-1.729×104T⋅Si2-9.438×1011Si3 As in , we use this active site parameterization in the framework of Eq. () to calculate nucleated crystal number:

nINP,X=∫log0.1μm∞{1-exp[-ns(T,Si)πD2]}nX(logD)dlogD. Hereafter, we refer to this formulation as the AIDA spectrum.

Figure  shows a comparison of the in situ crystal number measurements, taken from the NASA MACPEX (Mid-latitude Cirrus Properties Experiment) and the DOE SPARTICUS (Small Particles In Cirrus) aircraft campaigns. Data are used from the Video Ice Particle Sampler (VIPS) and two-dimensional stereo (2DS) probe during April 2011 of MACPEX and from the Forward Scattering Spectrometer Probe (FSSP) during January 2010 of SPARTICUS. Using simultaneous Meteorological Measurement System (MMS) pressure values, only Ni measurements taken within 20 hPa of the simulated pressure level of 232 hPa are used. Because the newly nucleated ice crystal number concentration is simulated, we use only Ni from the smallest size bins of each instrument (see caption of Fig. ). Finally, the same criterion for significant samples as in is employed: samples must continuously span at least 45 s. These MACPEX and SPARTICUS measurements, taken with shatter-resistant probes and analyzed with an inter-arrival time algorithm, are more reliable than older ones, especially for the smallest size bins that we consider ().

Simulated and measured Ni agree best for the PDA13 spectrum, followed by the PDA08 and then the AIDA spectra. The CNT spectrum overestimates the frequencies of Ni greater than about 10 L-1 by more than 1 order of magnitude and predicts no number concentrations less than 1 L-1. Measurements show, instead, that most of the smallest crystals occur at lower number concentrations, below about 5 L-1. The very high frequency of low Ni is missed by the other spectra as well, and all except PDA13 show slower decays in the frequency of high Ni than those in the measurements.

Model overestimate of high Ni at the coldest temperatures has been often noted (e.g., ). Along with this “ice nucleation puzzle” of low Ni at low temperature (), model–measurement discrepancy may be explained by in-cloud processes after nucleation: nucleated crystal number will tend to be higher than in-cloud crystal number, even when looking only at the smallest size bins. Preexisting ice crystals can inhibit ice nucleation (), while sedimentation can significantly reduce Ni. have termed the latter “sedimentation induced quenching of nucleation”, and found that omission of sedimentation by setting crystal fall speed to 0 m s-1 resulted in higher frequency of Ni greater than 1000 L-1.

Figure a through d show the annually averaged potential nucleated ice crystal number for each grid cell, given the vertical velocity and aerosol profile. The spatial variability in these fields is notable and reflects the large, documented spatial variability in INP concentrations (e.g., ). Including additional microphysics after nucleation will tend to reduce this spatial variability. Some common features are still observed between fields: over the Himalayas and Rockies, Ni is higher because orographic lifting generates stronger updrafts and more supersaturation; the Saharan and Gobi desert outflows enhance Ni,het; and for INP spectra considering black carbon (all except the AIDA spectrum), higher Ni,het occurs in regions of biomass burning (e.g., sub-Saharan Africa and the Amazon). In the Southern Hemisphere, especially over Antarctica, heterogeneous nucleation is rare, and Ni stays high because aerosol number concentrations are low and active site density decreases with temperature.

Elsewhere, Ni is highly variable and sensitive to the INP spectrum. For example, all spectra except PDA08 see higher crystal number in the Northern Hemisphere than the Southern Hemisphere. In agreement with previous studies, PDA08 predicts the lowest INP number, between 0.047 and 5.07 L-1, (Table ) and the highest maximum supersaturations (). When the input aerosol number is sufficiently high in the Northern Hemisphere, stronger competitive nucleation results in lower Ni, while the Southern Hemisphere remains dominated by homogeneous nucleation and higher Ni. The heterogeneously formed fraction field in Fig. S3 also illustrates these regions of competitive and homogeneous nucleation in the Northern Hemisphere (NH) and the Southern Hemisphere (SH), respectively. Updraft velocity and Ni are well-correlated; both have higher values around the equator for PDA08.

Compared to PDA08, PDA13 predicts about 1 order of magnitude higher INP number, between 0.57 and 28.6 L-1 and more frequent inhibition of homogeneous nucleation, as shown in Fig. S3, where the heterogeneously formed fraction of Ni is much higher. In localized regions of purely heterogeneous nucleation, however, PDA08 may still predict higher Ni. This can be understood in terms of an INP abundance, AINP≡NINP/Nlim, defined as the ratio of available INP to the limiting number to inhibit homogeneous nucleation. Nlim increases with decreasing maximum supersaturation, Nlim∝Si,max⁡/(Si,max⁡-1), and this increase in Nlim can outweigh the increase in INP number so that AINP actually decreases within PDA13.

Higher Ni in PDA08 can also be understood in terms of threshold supersaturations for nucleation, when calculated supersaturations are similar between PDA08 and PDA13. When these thresholds are less stringent, the competitive nucleation cusp of the INP-Ni trace becomes steeper and extends to lower Ni values. Where nucleation is competitive, then, as in PDA13 around the equator, very low Ni is possible.

Compared to PDA13, INP numbers in the CNT and AIDA spectra are about 10-fold higher, with median values of 50.38 and 52.51 L-1, respectively. High INP numbers result in almost purely heterogeneous nucleation everywhere for the CNT spectrum, as shown in Fig. S3c. The highest crystal numbers in any of the fields occur for this spectrum in Saharan outflows because of the high dust nucleation efficiency and the dependence on aerosol number concentration rather than surface area here. Large accumulation-mode dust numbers can yield large AINP. Ni is on the order of 1000 L-1 here, larger than any of the in situ measurements shown in Fig. . An overestimate of INP by CNT-based spectra has been reported elsewhere (e.g., ).

For the AIDA spectrum, mostly heterogeneous nucleation occurs in the Northern Hemisphere, while competitive nucleation occurs in the Southern Hemisphere. INP increases lead to frequent inhibition of homogeneous ice nucleation for these last two spectra. Again, higher Ni are due to higher AINP; here, the increase in Nlim with decreasing supersaturation is not enough to outweigh the higher INP numbers.

A final point can be made about the strong temperature dependence of the threshold supersaturation for homogeneous nucleation. Within the CNT spectrum, the heterogeneously formed fraction of Ni actually increases in the SH (Fig. S3) because at the coldest temperatures, the threshold supersaturation for homogeneous nucleation significantly increases, as shown in Fig. b. A fewer number of INP are needed to depress the supersaturation enough to inhibit homogeneous nucleation; the dust INP in the CNT simulations are efficient enough to shut down homogeneous nucleation.

We consider next which aerosol groups act as INP in the regions of purely heterogeneous nucleation. For PDA08 in Fig. a and c, both dust and black carbon play a role. Gradients in input temperature and BC contribution both appear around 40∘ S because the BC threshold supersaturation is a quadratic function of temperature in this spectrum (Fig. b) (). Below 60∘ S, the BC contribution is 40 % or higher for PDA08. This is unexpected because black carbon sources tend to be continental and anthropogenic, while land coverage and population density are lower in the SH.

For PDA13, dust is by far the primary contributor to Ni,het outside of a very localized region of deep convection around the Equator. The correlation for si,0,DM remains the same between PDA08 and PDA13 and decreases with decreasing temperature because observations show that nucleation on dust generally becomes more efficient at colder temperatures (e.g., ). PDA13 also uses an updated correlation for Si,0,BC, expressed in terms of surface polarity and organic coating:

Si,0BC=S̃i,0+δ01(FOC,FOC,0,FOC,1)×[1.2×Siw(T)-S̃i,0], where S̃i,0 is a baseline supersaturation of 30 %, δ01 is a cubic interpolation over organic coating, FOC, between lower and upper bounds of FOC,0 and FOC,1 (), and Siw is the saturation ratio of vapor with respect to ice at exact water saturation, since minimal nucleation has been observed at water-subsaturated conditions for heavily coated black carbon (). Surface polarity expresses hydrophilicity and is operationally defined as the number of water monolayers adsorbed to the aerosol surface at 50 % relative humidity, while the organic coating indicates the fraction of BC surface covered in insoluble organics. These parameters are source-dependent and difficult to determine, but this study assumes a high surface polarity of two monolayers and a low organic coating of 10 % to maximize any impact of black carbon (Table ). have also shown that these values describe aircraft engine combustion emissions, which would be relevant at this altitude.

The different aerosol contributing to INP concentrations, despite the same framework, can be understood by analyzing the expression for μX. Given that the same aerosol size and number distributions have been used in both runs (Table ), the difference is in the active site density parameterization. The observationally based terms making up the active site density are a threshold for water-subsaturated conditions, a threshold for warm sub-zero temperatures, a background aerosol number, and a baseline surface area mixing ratio (): nS,X=HX(Si,T)ξ(T)αXnINP,*ΩX,*. Between PDA08 and PDA13, the portion of aerosol belonging to the BC group, αBC, has increased by 3 %, while our input temperatures are too low for the warm sub-zero temperature threshold, ξ(T), to affect NINP calculations. The water-subsaturated threshold, HX, would completely suppress BC nucleation if FOC were taken to be 100 %; experimental evidence has shown that BC nucleation may only occur at water saturation when coating is significant (). But we have used FOC of 10 % and the threshold supersaturation has actually decreased for PDA13, as shown in Fig. b. These factors alone actually yield a higher active site density for BC than for dust.

The difference in contributions, then, is the result of changing baseline surface area mixing ratios, ΩX,*. A lower active site density is needed to obtain the same freezing fraction when ΩX,* is higher. Between PDA08 and PDA13, this parameter decreases 4-fold from 2×10-6 to 5×10-7 m2kg-1 for dust and increases about 3-fold from 1×10-7 to 2.7×10-7 m2kg-1 for BC. As a result, the freezing fraction of BC is much lower, even if nS,BC is somewhat higher. Dust becomes the primary INP for PDA13 because its freezing fraction has increased. Ni from PDA13 is lower in the NH because the large dust numbers there depress Ni,hom, as shown in Figs.  and S3.

Surface polarity and organic coating parameters are prescribed in these simulations and may be highly variable in the atmosphere. We have chosen a high polarity and low organic coating, so that BC contribution calculations represent an upper bound. For simulations with higher organic coatings, any INP contribution from BC disappears completely. But polarity and coating change with morphology and porosity, which change with source (). A more detailed consideration of the BC emissions inventory would be needed to more accurately determine these parameters and BC contribution to crystal number. Uncertainty also exists within the BC emissions inventory itself, and this, along with the coating and polarity parameters, will translate to uncertainty in the Ni field.

The sign and magnitude of the insoluble aerosol number sensitivities, ∂Ni/∂NINP, can be used to elucidate the active nucleation regime. Figure  gives an example with the annually averaged sensitivity of Ni to accumulation-mode dust number, ∂Ni/∂Ndust,a, for all spectra. In the Southern Hemisphere, sensitivities for PDA08 are of small magnitude (O(10-4)) and negative, as homogeneous nucleation dominates. There are localized regions of strong competitive nucleation in sub-Saharan Africa and northern South America, where sensitivities are of larger magnitude (O(10-3)) and negative. Sensitivities throughout most of the Northern Hemisphere are of moderate magnitude and negative, indicating weaker competitive nucleation.

The CNT field exhibits positive sensitivities throughout most of the Northern Hemisphere, delineated in white and indicating purely heterogeneous nucleation. PDA13 also contains regions of purely heterogeneous nucleation but around the Equator in regions of lower updraft and higher INP. When updraft velocity increases significantly – in the region of deep convection over Indonesia or over the Himalayas or Rockies due to orographic lifting – a sufficiently high supersaturation may be generated to exceed the threshold for homogeneous nucleation and induce competitive nucleation. For both the PDA13 and AIDA spectra, regions of large and negative sensitivities, or strong competitive nucleation, appear south of 60∘ S. INP numbers are considerably lower than Nlim here, but the threshold supersaturation for homogeneous nucleation has also increased at these cold temperatures.

The magnitude of negative sensitivities during competitive nucleation reflect the threshold conditions assigned to a given aerosol group. The lower the threshold supersaturation for an aerosol group, the more readily it nucleates and the more effectively it depletes water vapor; this corresponds to larger magnitude ∂Ni/∂Ndust,a before NINP surpasses Nlim and purely heterogeneous nucleation begins. PDA13 sensitivities to BC number are of larger magnitude than PDA08 values because Si,0,BC is lower for the polarity and FOC values used here. The cusp of the INP-Ni trace becomes steeper, and the competition for water vapor is stronger in this case.

∂Ni/∂Ndust,a is of large magnitude (O(10-2)) and positive for the AIDA spectrum due to larger predicted INP numbers. These sensitivities decrease in magnitude over the Antarctic because the active site density parameterization has a strong supersaturation dependence at cold temperatures (Fig. S5). If the temperature decreases by 5 K for a constant supersaturation, the active site density can drop by as much as 25 %. The effect of this active site density parameterization on Ni is discussed further in Sect. . ∂Ni/∂Ndust,a also decreases in magnitude over Indonesia because the large updrafts here generate enough supersaturation that competitive nucleation occurs often and reduces the annually averaged magnitude of ∂Ni/∂NINP.

Along with these spatial sensitivity patterns, we look at sensitivity time series without temporal averaging, which show the frequency of occurrence of different nucleation regimes. Infrequent but large magnitude sensitivities can have an important influence on the annual average (). Distributions of both accumulation-mode dust number sensitivities and input updraft velocities are presented at (2.9∘ S, 135∘ E) over Indonesia and (0.95∘ N, 64∘ W) over northern South America in Fig. . These points are denoted by diamonds in Fig. . Their annually averaged sensitivities differ significantly, despite their being in the same latitudinal band with similar aerosol loadings.

The location over Indonesia experiences high updraft more frequently, and the additional supersaturation generation translates to more competitive nucleation and larger magnitude sensitivities in PDA13, almost down to -0.1 LL-1. In PDA08, more supersaturation generation translates to more frequent homogeneous nucleation and smaller magnitude, less variable sensitivities, on the order of 10-3 L L-1. The location over South America has fewer instances of high updraft, so for PDA13, the system cannot always overcome the threshold supersaturation for homogeneous nucleation. Purely heterogeneous nucleation occurs more frequently: Fig. d has primarily positive sensitivities of small magnitude with an occasional large spike in ∂Ni/∂Ndust,a, which always corresponds to a large updraft. Relative to PDA13, PDA08 exhibits stronger water vapor competition: the peaks in Fig. c are about 4 times as large as those in Fig. a. This behavior can be understood in terms of a transition along the INP-Ni trace in Fig. a: Ni and ∂Ni/∂NINP respond differently to supersaturation generation based on how many INP the nucleation spectrum predicts.

The positive values of ∂Ni/∂NINP, for which nucleation is purely heterogeneous, can be understood as nucleation efficiencies: those aerosol which act as efficient INP generate a large increase in crystal number for a given increase in aerosol number. Rather than an inherent nucleation efficiency of a certain aerosol group, the sensitivity reflects an INP efficiency given the particular model state. Accumulation-mode dust has a mean efficiency of 0.0012 % (O(10-3 %)) in PDA08 and 0.079 % (O(0.1 %)) in PDA13, while coarse-mode dust has a mean efficiency of 0.61 % in PDA08 and 0.078 % in PDA13. AIDA calculates considerably higher mean efficiency of 1.4 % for the accumulation mode and 52 % for the coarse mode. Black carbon in PDA08 is 0.03 % efficient on average, 1 order of magnitude higher than the accumulation-mode dust. In PDA13, on the other hand, black carbon efficiency is an order lower than accumulation-mode dust and skewed toward lower values (not shown). Efficiency of organic aerosol is negligible, on the order of 10-5 % and skewed to values as low as 10-12 %.

From Eq. () during purely heterogeneous nucleation, ∂Ni∂nX=1-exp(-ns(Si,T)πD2). As the number of embryos per aerosol particle becomes large, the nucleation-active fraction of the aerosol population, which is equivalent to the positive aerosol number sensitivity or the nucleation efficiency, approaches unity. This occurs because the product of active site density and aerosol surface area becomes large enough that an ice embryo should always form on the INP surface. Shifts in the number sensitivities reflect changing contributions to Ni,het. To illustrate, Fig. a shows the distribution of a random sample of 5000 daily averaged dust number sensitivities, when ice nucleation is purely heterogeneous, i.e. ∂Ni/∂NINP>0. The coarse-mode dust number sensitivity is higher, and the accumulation-mode dust sensitivity is lower for PDA08 than PDA13 because BC nucleation has been suppressed in the latter. The active site density of PDA08 BC is larger than that of dust under certain conditions (Fig. S5), meaning that BC efficiencies are higher than the accumulation-mode dust efficiencies because aerosol diameter for the two groups is assumed to be the same. The coarse-mode sensitivities or efficiencies are even higher because their surface area is 2 orders of magnitude larger and outweighs a lower active site density.

The PDA08 distributions also have many more outliers because of the greater competition for water vapor between aerosol groups. The adjoint sensitivities are local in space and time, and in model grid cells without BC, dust in both modes is able to nucleate much more efficiently. In grid cells with more BC, the dust nucleation efficiency is significantly reduced because of the competition for water vapor between the two INP groups. The narrower range of AIDA efficiencies reinforces this point: this spectrum describes nucleation by dust in idealized conditions, and no other aerosols compete for water vapor. Its active site parameterization also contains no threshold functions that abruptly reduce nucleation. For application in global models, it may be more effective to use parameterizations from experiments with multiple nucleating aerosol types.

Once an aerosol population has reached its maximum active fraction or efficiency, Ni becomes less sensitive to the number of these aerosol. In PDA13, the coarse-mode dust population reaches an upper bound in its efficiency, and Ni sensitivity to coarse-mode number decreases to a value comparable to the accumulation-mode number. For low active fractions, Eq. () can be linearized so that fIN∼O(ns(Si,T)D2). Given that ns∼O(109 m-2) and D∼O(10-6 m) in the coarse mode (Fig. S5), the maximum active fraction is expected to be on the order of 10-3, which is indeed the value seen in Fig. .

Diameter sensitivities can also be understood in terms of nucleation regime. When nucleation is purely heterogeneous, diameter sensitivity is positive; increasing aerosol diameter increases crystal number because for a given active site density, more surface area increases the number of ice embryos per aerosol. During competitive nucleation, diameter sensitivity becomes negative, as more available surface area for heterogeneous nucleation reduces Ni from Ni,hom. As with number sensitivity, the magnitude of negative diameter sensitivities reflects how intensely a certain aerosol group can deplete water vapor. The magnitude of positive diameter sensitivities is larger for coarse mode than accumulation-mode dust in all spectra (Fig. ); an incremental increase in diameter generates more surface area for larger particles than for smaller particles.

The magnitude of positive diameter sensitivities also reflect active site density. From Eq. (), during purely heterogeneous nucleation, ∂Ni∂D=2πDnXns(Si,T)exp(-ns(Si,T)πD2) which shows that ∂Ni/∂D∝Dexp(-D2) and ∂Ni/∂D∝nsexp(-ns). The magnitude of diameter sensitivity first increases, then decreases, with diameter. The larger the diameter, the faster the sensitivity decreases after its maximum and the larger that maximum sensitivity. Again for active site density, the magnitude of diameter sensitivity first increases then decreases with ns. And the larger the active site density, the faster the sensitivity decreases after reaching its maximum value. The first effect is stronger because ∂Ni/∂D is proportional to active site density but to the square of diameter.

Figure b is constructed again from a random sample of 5000 daily averaged dust diameter sensitivities in the purely heterogeneous regime, i.e. ∂Ni/∂DINP>0. The maximum coarse-mode diameter sensitivity is smaller than that of the accumulation-mode diameter sensitivity for PDA13 and CNT because the higher number of accumulation-mode dust particles outweighs the larger coarse-mode surface area. The AIDA and PDA13 spectra tend to reach the same maximum diameter sensitivities (10-11 µm cm-3 in the coarse mode) as both have reached their maximum active fraction. These features do not characterize the PDA08 distributions because of competition for water vapor with black carbon. Given the higher active site density and equal surface area of black carbon relative to accumulation-mode dust, ∂Ni/∂Ddust,a is smaller than in the other spectra. The surface area increase from the addition of a coarse-mode dust particle outweighs the higher BC active site density and ∂Ni/∂Ddust,c in PDA08 is comparable to the values in the other spectra. In summary, spectra with large active site densities will be highly sensitive to aerosol diameter over a limited range of these diameters, while spectra with lower active site densities will be less sensitive to aerosol diameter but over a larger range of these diameters. These trends may be convoluted by competition for water vapor with other aerosol species.

The above discussion has focused on insoluble aerosol sensitivities. Soluble aerosol sensitivities, ∂Ni/∂Nsulf, are always positive because the addition of these soluble particles enhances homogeneous nucleation and crystal number, regardless of the insoluble INP profile. When purely heterogeneous nucleation occurs, ∂Ni/∂Nsulf is zero. Sulfate sensitivities are generally on the order of 0.001 cm3cm-3 but can be as large as 0.025 cm3cm-3 at the coldest temperatures in the SH. This field does not change in magnitude between spectra because the treatment of homogeneous nucleation is identical in all cases. ∂Ni/∂Nsulf is smaller and less influential than the updraft sensitivity fields, similar to the findings of , for which the aerosol size distribution did not strongly affect the number of nucleated ice crystals.

Temperature sensitivities, ∂Ni/∂T, are generally negative because colder temperatures tend to facilitate ice nucleation. An increase in temperature may exceed the threshold temperature for a certain aerosol group, deactivating it, and allowing homogeneous nucleation to generate a larger Ni. This phenomenon can be observed in both the PDA08 and PDA13 fields, in which positive sensitivities fall exclusively at the outflow of Saharan dust around the equator where input temperature is between 225 and 230 K. These temperatures are in the range at which the water-subsaturated threshold function for dust drops (T0DM=-40 ∘C and ΔT=5 ∘C), so that the primary contributor to heterogeneous nucleation depletes less water vapor and homogeneous nucleation yields higher Ni.

The magnitude of ∂Ni/∂T is smaller than expected from classical nucleation theory, probably due to counterbalancing effects. For example, as temperature increases so does water vapor diffusivity, which enhances crystal growth and reduces crystal number. But latent heat of sublimation also increases as temperature drops, which slows the crystal growth rate. The homogeneous nucleation coefficient increases by 1 order of magnitude with only a 30 K drop in temperature (). The threshold supersaturation for dust, however, also goes down, so that deposition nucleation can more easily inhibit homogeneous nucleation. These various temperature dependencies may cancel out and lead to lower temperature sensitivities within the model. have noted an intermediate regime in nucleation experiments for which ns isolines are independent of temperature and change primarily with supersaturation. Similar compensating effects, which cause low temperature sensitivity in the parameterization runs, might also explain this experimentally observed, temperature-independent regime.